% Define a LTI system and then compute the 
% transition matrix Ph_s and the transfere function G_s

%%%%%%%%%%%%%%%%%%%%
%%% State-space representation %%%
%%% x_dot=Ax+Bu, y=Cx+Du  %%%%
%%%%%%%%%%%%%%%%%%%%

clear all
close all
clc

% define the matrixes A, B,C and D of the ss representation
A=[-2 -1.5; 2 0];
B=[2;0];
C=[0.5 0.5];
D=0;

% define a symbolic variable for the laplace variable
syms s

% define the identity matrix
I=eye(2);

% define Ph_s
Phs=inv(s*I-A);

% define G_s
Gs=C*Phs*B+D;

% define initial condition x0
x0=[1,2]';

% define X_free(s) for x0
Xs_free=Phs*x0;

% define Y_free(s) for x0
Ys_free=C*Xs_free;

%compute the response to step signal (U_s=1/s)
Ys_step=Gs*1/s;
Ys_step=simplify(Ys_step); % simplified version of Ys_step
% display a simplified version of Ys_step
disp('Ys_step=');
disp(Ys_step);

% compute the antitrasform functions

xt_free=ilaplace(Xs_free);

yt_free=ilaplace(Ys_free);

yt_step=ilaplace(Ys_step);